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Thin set

A subset $ A $
of a domain $ D \subset \mathbf C ^ {k} $
such that, for each point $ z \in D $,
there exists an open polydisc $ \Delta ( z, r) \subset D $
and a function $ f $
which is holomorphic, not identically equal to zero, but which vanishes on $ A \cap \Delta ( z, r) $.